Chiral anomaly

For simplicity VR=V, k=O .

eiwttl.fm#eiSi4IDa4rdex=wnstfo4royIoqoyieiKi4IDta4r+i4T04)dex

= onstfoy out eif ( i # 4 + IFAPR 4) atx

where PR = ¥5 a the projection to R . handed spines .

Under ( the regularized version of ) this definition of W ( A ], few f A ]

actually has an A- linear term .

idde WCAH.

= ifdax Are laid S ftp.lf#oti/daxfAr.EkxsofWafAf- I

.

Lorentz → := if oex dofdx) ( i(4V"TPR4) 6) if day if# Pr 4) ( b ) )um

= ifdxd.FM/d4yt4tyxog(vnTaPr4(x#)sAPnYlsFT4

)

- JA ( s ) = dyutbe 'i95

⇒Jdaxoredxifdye

" "G) ty•g(VrTaPryFys)VTbPr4Fikl )

""

gy¥µieh←I go;÷µieiwtItx *

do the g- integral ⇒ Raj 451-9+4 . he ) =) hehh

,= hear

¥4 fax

%ea.ie#tiiitruCTaTs)fftahatglrPr*VPrtf

)9

quadmtkally divergent .

Paul :- Villars peg : to take Care of the quadratic divergence

,

introduce 3 regulator fields.

0 1 2 3

mass 0=10 A 12 ^3

gtatfermi bose fermi bose

( Eo = l ) ⇐,=

-1) ( Eel ) ( Ep - i )

with ¥.

Ein .?= 0.

ffatgfatrslrnpr

# VPr*t )

→ Jaffa El

ftp.#VPE)=:IwCe)

11trsftp.h#kvup..tetA=)Chey

'

. n' ← it like it

go :Sa→SLi

Prn ..pro#_trs(MPr(htETVPrk)Khteknitie)( htnitic )

Trs Pr = Trs = I tr,

I = I = 2

Trs Par 'T= frfr 'T = ¥ tryin = 22 "

trs Prr"

ru re r'

= first"

V re 8 't I tyre r 'T re y"

= 2 ( n'"

n"

- 449 "+ y

'hey "- i Emea )

I" H )

¥fgI÷µ ¥

.

⇐21445 h

"- chill . H

" I Kees" W - ie

" " " "

q .

h, .

)

(( he91'

- hit if ) Ch '- hit it )

;

¥t , f - 2h " E- 499 ( II eilognixlgl - i I - E )+ n

" ( 6¥,

ein ignite ) )

i . id de W CAT to = fdex Eau , e- i "

truttaTb ) i 9,

I' "

(9)-

*

JEW CAI has A - linear term.

0

However,

this can be cancelled by adding a local counter term

to W CA ].

Let's consider

OWIAI =/ dax tru ( Ar C Ch"

+ D2" d) Au ]

.

Then,

de OWCA I = 2 fix tru f Dre ( Ch"

+ DT" d) Au )

.

iddfow CAT ! = 2 if dextro ( CI 't Dn" d) SA, ]

MA,

= sit be' iex

I2ifd9xxe×truth Tb) C CT "tDn" foil )

Eam E" "

( i 9,

)

id dew CAT lot iddfow CAT fo = o

⇒ 3¥ ,16 Eiht log hit E) + 2K - D 94=0

.

We may also add local counter term

I WCA ] = E) da x Tru fAn ( M"

d'

- d " du ) An ),

which is gauge invariant at this order : fore O' WCA ] to = o.

We shall consider

W' CA ] = W LA ] t OW CA ] t D W CA )

for the above C er D ( C = ftp.EC-inilghi .D= 6¥

,. )

and E = Eh # p to be determined.

Now let us compute of Idf W'LA ) for W' = W -10W -18W,

C- = EXIT ?

I A = DNT'

e- i E "

, 82A = DNT' e-

i " "

.

idvdewtaslo-ifaxD.eacxsdd.SI?I!#: ⑧

+pile'

Coffield"dit④" " )

.

④ :id.dgwn.IT?/o--e- " "

trutta -17ft 'Ve ) -1 zilch"

- Dl" -

i )q

I

,A=dKTbEis "

- Zi

E ( n' "

of - gyu , )

= j2 truth ) Cn'"

EEE ) ( Eieinibgnixlgtey + E)EYTThe

aboveC

,D and E = : Ekka

,a

For ⑧ ,

addition of 0W -10W has no effect.

⑨ = ( ifaax i ④DE Pr4) ex , ifdy if# Pr4) G) ifdzilIHPY.is?n.

qI EM = EH -19

,d. AH =d5TbE

i ", face ,

-

- dzetceit )

= ibfdxd ,E C x , / day e-

" 99Jdaz e-

it Z

←D tf ( Y" -19%4×54519) VTbprtsTTHVTPY.TK ) t Cubs ) # leet )

=i%axo"' " "

fd¥qfHy*sfrMPrn¥Vt'Re HT 'Re ) t ( THE)Eh , e-

i " " ' "

i Cpt 'd,

= - i" ) dax Eun

" " " "

Tru LT 'T 'T '

)fY÷µ Trs ftp.#Pr*VPr*trePr # )-

+ ( 9. able , ( t.ae )

f Pauli - Villars

.

C-igck.li )

where

gch ,m ) try I ¥91 PR h¥VPR¥m NPR h¥m )

um

Use Hat Pr = ( KI - He ) Prum

f V"

Pr = RT"

= packet ) - (k-py PR

= Pal # - on ) - ( ¥ - m ) PR t m ( PL - PR )

9th .

m ) = tr f RV Pr # NPR ¥m - Pr¥mVPr¥ePR

+ mlk - Pal ¥7 NPR # NPR h¥ ]= Trs

ftp.htymreprh#.m-h*ImruPrhImVePR

t m(R-pn)Ck#tnlRVhHPr(h#-m#*9km 't it ) ( bi - m 't it ) ( ( h - pfm 't it ) ]

(= m f - PrCh# + m Pa ) WK re PRC htt tm )

= my - Cheat Where Pr + VKrepnch.pt }

= - m'

CPI r-

K re Pr

Using this,

we find

§Y÷µ.

Eiglh ,hi ) = IM -

P ) - I"

( E ) -1 J"

I 9. p )

where

IT e) = fY¥p,E. Ei trs ( Mk WPR *¥ )

as defined earlier,

and

549. P ) -=/ ⇐ I . ni ,

trs CPH Where Pr )

*4'

- hit it ) ( hi - hit it ) ( th - PT - hit it )-

After some computation,

we find ( in the limit 9 ? P - 9,

P'

e hi ) :

-1491 = seize funniest eibnitlgc.it . E )

+ y" ( f ¥

.

Ei nilgai + of ) ) caseated earlier )

2 2

Ju 'll. e) =zi I

unite're ) - Cree

#in ) ) - ¥⇒E"

" " '

4. Pm

:JY¥,fish . hi ) = ,¥,{ icvoi-oigel.EE. lognitlogfi) - I )

- ifguep'

e pipe ) ( Eilghitbgtp 't - E )- em " "

Ep . Pin )

i. ⑧ = Jdax Ecx , Ei " " " "

. jffpx {irrutiltb.TN/uueai-oiee).EeilosniHgfit- E )

- tie peeps ) l eilgnitigtey - E ) )

- i tru #II. Ty ) E" " " '

9. Pm )

④

ifdxlohh-r.ETdkldidgwaf.IT/o--fd4xfd!TcE" "

,ENT ' ]d

× truth )U" Ee ''THE.

lgni-ilogtit.EE/eisx--fdxEcxiii" " " "

RICHEST'

) I n' E- eeq-

)

"

-0truths ,

Ty )

× fEi Eibshixlyfi ) . f. E)

If we set - It E = - I ( ie .

E= - I ) ,

only the E" " "

Er. Pm

term from ⑧ remains in i did, de W' CAIL

..

That is,

iv. de W' CAI != Idk EH , e'

" " " "

j÷ .truth .tl/E'

" " " '

4. Pm

for G- = Eck , Ta,

d. A = DNT'

e- i 9 "

, cha = DNT' e-

it ?

On the other hand,

for

Ae IA ) =/ tru ( Ed ( Adv -12A '

I ),

iv. detail. =) trufedldiADKA-RAdd.AT)

=) tru EMT 'd ( E " "

Fdn-

df E'' " "

Tune ) ) + Clay-

e- ipkf - i Pindar )T'dH

-E

" "

fi 9. die" ) Tbdx

'

E i " "

f- iPads ) -19dm

=/ Ecu , Ei " " " "

Tru ( T 'T 'T '

) 9. P,

du " dkudkt.dk ' + Cle . 2)-

- E" 'M d4x

=fdanEcx, e- " " " "

truth -17 4) E' " '

9nF .

match !

i. idea of hi CAIL.

=

iddac.CAT/odeWIAI=AelA

I holds at OCA '

).

-

Summary By computing in & -

\we have seen

few = f -1¥ Etr @And A ) + OCA' ) o - higher

JEW = Juiz Tru @DANDA ) + OCA' ) o - higher

modulo defend ),

consistent with AECAT = f fat E

trftanfa)

AREIA ] = faith ( E dl ADA '- IA 'D.

-Wemay compute

hand higher

✓ n

to fix 0 ( A'

) or higher order terms.

But that is not necessary ,if we use

general features of anomalies .

.

Generalfealuresofauomalies

① t-noIWAIAAisbcd.ie .

{ dax ( Polynomial of finitely many derivatives of Ea Ap),because it comes from regularization procedure, which is

non- trivial only for divergent diagrams c→ local .

② ( Already discussed )There is a freedom to modify WCA) by local counterterms of A . Thus ac.CA) is

dhhedonlymodulodellocdfunctionalof-AD.TOJefe , WCA] - de. fewCAT = Ice. . WCA?

Thus,the anomaly must satisfy

dezae.CAI-te.AEIAJ-acez.eduwess.EU#noconsisencyConditin .

Note : Trivial part de local CAT satisfies itfor a trivial reason .

④ Ae LA ] = const ) tru (Ed(AdAtfA' ) )satisfies WZ consistency condition . ( exercise )

⑤ It is also the unique solution ( of course

modulo trivial terms of local CA ) ) with the

" initial condition "

AELA ) = const) tf ( C- DANDA ) + OCA') on higher .Thus

, our computation ( from # or -0- )is enough to prove

aeRCAI-fzaia.tn/EdlAdAtztA3 ) ),

and hence also

a (A) =/ C- trltanfa )

by the rein of AILA ) & Aco.

(A. O) for Gx UNA.

Alternative : Fujikawa's method#

xo= - ix4We work in the Euclidean signature

To = - it", { y".ru ) = - 28

""

µ,u=i, - -- 4

Tg = ivory -y'= yay 'V'r ' = - y ' r'y ' y'

,

V5= ids ,KV"= - V''Ts

e.g . S=f4, yr=( go.to

"

) O''

= - F Pauli 's a ii. 2.3

04=84 = - i Iz

Trs Tg TM'. -

- yMm - i

= o

tyre = trsrs V'T"= o

tyrr"Vver- = - scene- ( e' 234=1 )

L = i Fr"D, 4

= i4r°Do4 tilt:D,. 4

" LE = - i IV"D,-4 = - i -44'D, + VID: ) 4

Do- i Dc,

AxialanomalyDirac fermion 4 in a representation V of G .

e- WE(A)

= Joy out ei IT#4 d"= det ( i Da )

e-WE LA .A' ]= Joy go, eif THAT Mrs ASI

) 4 dax

For AS = idE

e- WELA , ide]

= g of eif 4-( BA tr" Vs ionE ) 4 d"X

-

4-eith Dfa e''Ere y- -

UT' y'

=) @( e- icky') @( I, e-iers) @if I' Da 4 'dax

=/ (det eick )'

Dyigy, eif I'

#4 'e'X

= (det eick )'

e-WECA )

or dat Leith ix. EEK) =@teams) . dot fi Da ).

-

'

' - 8¥ WE LA ) = Tr ( Li Ers ) . - i - divergent .

Need regularization .

Note : If Da has eigenvalues { tin 3nF , ,det ( i a) = II ish . C - ish) = exp (⇐ login )and it is divergent . But it may be regularized by

exp( E. e'''""

login ) = exp log Kanye""")

= dat ( i#e''#m)

,

Then detf@iErsipaeier.yeYeitbDn.e 4)YazII

get fevers E#m) det ( ikat

""

)-

'

i- d WE LA ) = Tr ( zier, e-

DAY" )= In@ ,

2.'

EVs e- DAY" Ya ) (4.3 basis

= fix÷,e-

'

'↳

trashier, e- DAH)eik×

used the plane wave basis.

e-ihx

aeihx = T " ( ik, t dat Ar)

e-inDa' ein =p"y- ( ihre dat Ar ) ( ihutdut Au )={Er".ru > +fly".ru} ( dat Ah , dueAT-- = Fru- fu T: you

= - d"-

( ihre data, ) (ihr toutAu ) + IV" Fm

=h2-2atAAf + fr" FuY

e-in

e- Rati eihx = e- WH- X

; X = Yw c- EV"

Fuhr

= e-Mr x"

Ehxtrv.osr.ee#meihx=e-hYmtru*lrsE.oen7xn ) . I

need at least ( IV"

Frfr )?

= e-""Nu.dk/titnr'

"

Fru)'t . . . . } ) - I

- . . . = sum of powers of ¥m, ! ("F

"

ret.

l

with me l Z l in various orders .

so÷*e%=¥÷.J dado, e-Mm ha. . . hire ~ httpCTI )

⇒ Mtl Z1 tem → o as A→ co

-: - d WE LA ] = Tr ( Zier, e- DEH )

= 2i/dxECxsJE÷, e-"" trues ( rst " )= zifdx tix, ¥, trslrsr"H") truth. Fga )-

- 4 C-'" P"

= 2 i faxx E Cx, z⇐÷, E"" Nut Fou Fea )

= if C-HI -2¥, Tru (FanFA ).

i W CA ] = - WE CA ]

DE W CA ) = J E I÷, trutta nFa )

Chiral anomaly 4,2 in representation V of G.

e-WEH '

=

"

fgqzgtyz ei/4IBA%d× "

= Const foyaiynoqoq IKE# 4h + EOK )dy

= Const / 04dg ei 14T ¢ + * Pr ) 4 dax

*A ,R

= deti Bar

Da ,r= 0kt Bapr AIL 012+5'

#SPR

= ( Prt PLF) Ban ( Put SPR )

EWEH' ]

= det¢Pr+R5'

) i Dan ( Pe 'S Pr ) )

= detl( Patti'

) ( Pas Pa ) ) . deti Ban

= det ( Prs + R 5'

) . [WECAI

÷e← - 11

It BE + ...

:. - Jewels ] = Tv ( VSE )

.

..

.

divergent .

regularized Version

- Bahne

e-WEH '

= deti Bar

E Daintye-

WEH ' ]= deti Das

,r

- Haw e- Bite= det ( terse )E

. det i Dar

a - de WECA ] = Tr ( me E#£h '

).

BE = ( Pr - Pc ) E

{ c-Da ,n=PR8k+R #Pr

Rain = PRO Da Path # DPC

= TrlPre e→#h '

- Ree' # % '

)

= } Tr E ( e→#tie * day + } T.me ( Edt #the # any

⇒

!2! = tfdxfgelaegeihktrosers ( e→# the# My eihk

! exercise

= fax II. EMT'

tru ( Ed ,d Aude An + EAUA. Aa )) .

D= Efdtxfgfead, , Eihkty

. ,

E ( e→# the # 8h '

) eihe

!,

exercise

= fax qgttre ( Gid . At J d. At 2 And "oA - 28"

d.A) An

- And 'Aa*Jan . A"

+ d"

( An A. A - AH, .

Au + A. A Ar )]exercise

= fefdclxgtfa,tr| -3A A. A + And 'A,

- I And "d ' A - AA d. A

- 2A" JA

, .Au - at Arau At '

A" ]

,

ni - SEWECA ] = ftp.tru Ed ( ADA + EA'

) + de be [ A ]"

idew [ A ]

.: ofwtt ] = fate true dl ADATEA

')

mod trivial terms 8c-

Ioc [ A ]. ¢

Note Axial anomaly in any even dinar D= 2h.

✓ - matrices in D= 2h.

{ V"

,V 4=-28 " "

pro -

-I

,- -

,

d

da

represented on See Q2did )

( V'

.

.Vd )

'

= fi )-

d

You ,:= i r

'. - rd : tail

, Vat ,TE - r

"

Vat ,

trdra.ir" '. - r

"

I - til" IKE " " "

e.* yaex .tn

reg

def ( eitr "i

a

" " I= def e' ith - "

t

def ( iDAY- Sd We LAI = Tr ( 2iEVa+ ,

e-¥ "

)

=

2ifddxf.ge#peiktru*fera+ie-Dh-Yi)eih

"

÷ easy , exercise

= 2 if Entity Fa)"

Axial anomaly and index theorem

If 11 Fall to at 1×1 - a fast enough ,we may view A

as a gauge potential on Sd= Rdu @ ),

and we may consider

the Dirac operator DA on Spinors on S&.

( To be precise spinon with values in a Vector bundle E

with fiber V on Sol. )

fgyog eissd 5*4 " dO( 4. F) = ZOCA ]

Zola ,ide ] = foyout eissa EE

' " # EYEdue

due,

y'

y.

=/ @ fierce.yjqyeieva.ci ) ei)sd4" # 4 'dM0( e-iera # µ

,gieitruy

To see the Jacobian,

let as look at DYDI.

It Can be defined using th mode expansion of 4 & 4-

with respect to the Dirac operator

Da : SCE ) - SCE ) ; SCE ) = Else .Eos )

" 11 the space of SpirorsSRIE) SHE )

to * FeesICE )

DA is an elliptic operator - - - Ker # is finite dimensional.

Let { why CKERDANSRE )

{

GETC KerBanq⇐ ,

be basis .

Also the eigenvectors QF of Ba with non-zero eigenvdus IX,

is of the form 4n±=qR±ei where PNRESRE ),

4ie£CE )

and $akR=xn4i , $a4t- HE

Then, 19041¥ Yeah < SR⇐ )

basis

{

etyseniuceil

!,

CEE )

4r=EE both + I been" .k=¥Eboi%i+IIbi4i

Fr = ,⇐5oIeoFttEbieit,E=÷E5areoYtuEbEeY '

I

& DYJF =¥IdbYid5oY¥Idbgtd5y II. db 'idbid5Ydbe

Under the axial nation ( with Constante ),

eieratiyr = eicyr eictt ' 4<=5*4

Freier 't '= Free 4-Leicht = Tyeit

the modes transform as

b.? → eieb !,

b ?-

E '⇐b ?

5.4 - eitbr,

5. is it bit

.

'

. O ( e '

' era ' 4) g ( fEitrdt ' )

= FE dleitbo ? )d( Eitbo? ) Et dleitblg. ) dleitbg? )

× It dliiebnydleiebijdcetbnrjdleiebi )

dl A b) = A- ' db

=%iej2nR(E

it )m . 84 OF.

= die ( nr - no )D4 OF

.

Comparing with - date'

WCA ) = Zitynifenttru #Fay,

Sdwe have

nr - he =

Jganttretita)

"

.

This is an example of Atiyeh - Singer index formula.

nr - nee dinfkerDa NSRLE )) - dim(kerDanSdE) )

⇒Xa :S, # →SDED Ker #a

:SdE ) -5,2kt )112

Coker # a:S

.DE#lED=dimKer$a:SrLEtflED-dimGker(D/a:SnlEtSdE) )

= index ( Da :

SRCEI- ECE ) )

.

The index formula for a Vector bundle E on an even - dimensional

8pm manifold X is

index #a:S ,zKt - EE )) = §CHCEIAT

,

where ch€ ) = tre(e¥FA ) Chern character of E

A×

= I - tap ,( × ) + -.

- A. roof genus of X

÷ ' true power series

of Pontryagn classes Pjlx ) of X.

Pontryagin classes =o for X= Sd ( as IRI'=TS' to B )hence ffgd = 1

.Thus the index formula for S

"is

index #:S,

# → HE ) ) =

fgnchl E)

=

Spirent EeFat )

which is what we've seen from axial anomaly via

Fujikawa 's method .